I am concerned that the defenders of full-blooded second-order logic
(SOL) are missing an at least plausibly defensive position: namely
that while Quine's `set-theory-in-sheep's-clothing' argument provides
a case against full-blooded SOL qua the logic of sets (considered in
some sense to be an extension of ZF set theory), it still leaves open
the possibility that we can regard full-blooded SOL to be a logic of
properties: that is, we take the position that the second order
quantifier quantifies over properties of sets, and also that this is
not equivalent to quantifying over sets of sets. I seem to recall that
George Boolos took this position.
If one accepts the cumulative hierarchy as the guiding intuition
behind ZF set theory, it seems to me rather strange to regard ZF set
theory as an axiomatisation of arbitrary properties of sets.
Another interesting position arises if we reject excluded middle (PEM)
for propositions involving second-order quantifiers. The
inconsistency that arises in the Calculus of Constructions (an
extension of Girard's system F) when we add PEM is worth thinking
about in this context.
Charles Stewart, Boston University
http://achilles.bu.edu/cas